Webbed Space
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a webbed space is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
designed with the goal of allowing the results of the
open mapping theorem Open mapping theorem may refer to: * Open mapping theorem (functional analysis) (also known as the Banach–Schauder theorem), states that a surjective continuous linear transformation of a Banach space ''X'' onto a Banach space ''Y'' is an open ma ...
and the closed graph theorem to hold for a wider class of
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
s whose codomains are webbed spaces. A space is called webbed if there exists a collection of
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s, called a ''web'' that satisfies certain properties. Webs were first investigated by de Wilde.


Web

Let X be a Hausdorff
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. A is a stratified collection of disks satisfying the following absorbency and convergence requirements. # Stratum 1: The first stratum must consist of a sequence D_, D_, D_, \ldots of disks in X such that their union \bigcup_ D_i absorbs X. # Stratum 2: For each disk D_i in the first stratum, there must exists a sequence D_, D_, D_, \ldots of disks in X such that for every D_i: D_ \subseteq \left(\tfrac\right) D_i \quad \text j and \cup_ D_ absorbs D_i. The sets \left(D_\right)_ will form the second stratum. # Stratum 3: To each disk D_ in the second stratum, assign another sequence D_, D_, D_, \ldots of disks in X satisfying analogously defined properties; explicitly, this means that for every D_: D_ \subseteq \left(\tfrac\right) D_ \quad \text k and \cup_ D_ absorbs D_. The sets \left(D_\right)_ form the third stratum. Continue this process to define strata 4, 5, \ldots. That is, use induction to define stratum n + 1 in terms of stratum n. A is a sequence of disks, with the first disk being selected from the first stratum, say D_i, and the second being selected from the sequence that was associated with D_i, and so on. We also require that if a sequence of vectors (x_n) is selected from a strand (with x_1 belonging to the first disk in the strand, x_2 belonging to the second, and so on) then the series \sum_^ x_n converges. A Hausdorff
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
on which a web can be defined is called a .


Examples and sufficient conditions

All of the following spaces are webbed: *
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to ...
s. * Projective limits and
inductive limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categ ...
s of sequences of webbed spaces. * A sequentially closed vector subspace of a webbed space. * Countable products of webbed spaces. * A Hausdorff quotient of a webbed space. * The image of a webbed space under a
sequentially continuous In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
linear map if that image is Hausdorff. * The bornologification of a webbed space. * The continuous dual space of a metrizable locally convex space endowed with the strong dual topology is webbed. * If X is the strict inductive limit of a denumerable family of locally convex metrizable spaces, then the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...
of X with the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the t ...
is webbed. ** So in particular, the strong duals of locally convex
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a Metric (mathematics), metr ...
s are webbed. * If X is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.


Theorems

If the spaces are not locally convex, then there is a notion of web where the requirement of being a disk is replaced by the requirement of being balanced. For such a notion of web we have the following results:


See also

* * * * * * * * * * * *


Citations


References

* * * * * * {{Topological vector spaces Functional analysis Topological vector spaces